Friday, 11 December 2009

A mathematical physicist friend of mine once said that he looks forward to the day when physicists will produce a Grand Unified Theory (GUT) that will consolidate strong nuclear force, weak nuclear force and electromagnetism into a single interaction. This would be a stepping stone to the creation of a Theory of Everything that would also assimilate gravitation and thus unite all the strands of modern physics.

I was struck by his certainty that such a theory is possible. He seemed to be making a scientific prediction based on an aesthetic sensibility - unified theories are more beautiful therefore a unified theory is correct.

Unification is certainly an important part of the progress of a science. Occam's razor is a well-established principle for judging the utility of a theory. Entia non sunt multiplicanda praeter necessitatem - entities are not to be multiplied more than is necessary. In other words, the simplest theory that fits the data is the best.

When Sadi Carnot showed the equivalence of heat and mechanical work it was a victory for physics because physicists could now explain natural phenomena using less entities. At a superficial level, his discovery was useful because students now had to tax their brains with less concepts in order to understand both heat and motion.

But Carnot's unification had also produced a theory was more correct than earlier theories that thought of heat as a substance called "caloric". The mechanical theory of heat turned out to explain more phenomena than Carnot had originally considered. The laws of thermodynamics could not have been formulated without Carnot's insight.

Why should the simpler theory prove more correct?

Marcus Hutter believes that understanding is fundamentally an act of mental unification. The Hutter prize offers a reward for anyone able to produce a better compression of Wikipedia. A high compression ratio requires a deep understanding of the corpus - in this case a snapshot of human knowledge. If unification is in some sense equivalent to understanding then a unified theory is more likely to be correct because it is a better approximation of the phenomena in question.

However, it is a leap of blind optimism to assume that a G.U.T. is possible just because if it existed it would be useful, beautiful and likely to yield futher insight. Desirability does not imply feasibility.

In The Mythical Man Month Fred Brooks draws a distinction between essential and accidental complexity in software systems that is very pertinent to the possibility of a G.U.T.

Accidental complexity is caused by defficiencies in the solution. This kind of complexity can be eliminated by improving the approach to the problem. I would argue that the seperation of heat and mechanics was an example of accidental complexity caused by a lack of understanding of the nature of heat. The mechanical theory of heat was a successful simplification because it removed complexity that was never part of the phenomena itself.

Essential complexity, on the other hand, is inherent in the problem. It is impossible to build a solution that is less complex than the problem it is designed to solve.

The universe, like any other corpus, has an uncomputable Kolmogorov complexity that limits how simple a correct theory of physics can be. Though we cannot ever know the essential complexity of the universe, it does have one. There is an unknown and absolute limit to the unifying efforts of physics, so we cannot ever be sure that further unification will be possible.

Perhaps physics will encounter new phenomena that require new multiplication of entities to explain. Perhaps we are close to the limit and though we might incrementally simplify our theories we will never be able to reduce physics to less than four fundamental interactions.

We cannot hope to make out theories more unified than the phenomena they describe and still hope to make them correct. As Albert Einstein said (my italics):